There is, of course, Kahn Academy for fundamentals. We have already had a discussion on How to learn math.
What resources exist detailing which mathematics to learn in what order? What resources exist that explain the utility of different mathematical subfields for the purpose of directing studies?
A few points with pure math in mind (notes to my past self, perhaps):
Don't shy away from revisiting elementary material. It's not there just as a stepping stone to more advanced material, to be forgotten once you're through, it should remain comfortably familiar in itself.
This is in regards to low-level math (the higher maths are beyond me), but I thought some of you might find my story inspiring:
I was never fond of math. I got through Calculus in HS, but literally spent most of my math classes sitting in the back coloring in rainbows on graph paper. After getting a BA in history though, I decided I actually wanted a useful degree, and decided to get a second one in engineering. Of course, first thing you had to do was take a Math Placement Test.
It had been over 8 years since I took my last math class, which had been "Math for Elementary Teachers", and I had pretty much forgotten everything past very basic algebra...I could remember the quadratic equation, because my teacher had taught it to the tune of "Pop Goes the Weasel", but I couldn't remember what it did, or what a,b, or c was supposed to represent.
Anyway, NOT being willing to pay thousands of dollars, and take a bunch of boring intro classes to work my way back up to mathematical literacy, I decided to spend a summer reviewing math on my own. I got a bunch of "For Dummies" and "Demystified" books from the library, and started with basic arithmetic and Algebra 1. Over the course of the summer I worked my way back through everything I learned in jr. high and high school, and managed to test into Calculus (as high as the math placement test would go), saving myself thousands of dollars.
I have never learned anything so well, as the I did during review work I did that summer. Those 3 months of self-motivated study are probably the best investment I ever made in my learning. In high school, I only ever understood one concept at a time, which I promptly forgot after the test. Studying them in one fell swoop allowed me to understand it all as a whole.
From then on, with a firm foundation from which to build, math seemed easy. (well, except for Calc 3, but that's a whole nother story...)
I'll admit that its more like algebra-to-half-an-engineering-degree. I got divorced a while back and don't have the resources to finish grad school.
Also, I would say that it's not so much that I was ever inherently bad at math. I never had to study at it or anything. I personally just think I was socialized to not like it.
I've read somewhere (but of course I can't find it now, as usual) that when males standardize test really high in math, they are more likely to be worse at language arts. However when females test really high in math, they are more likely to test even better at language arts.
I know this was the case for me. Math was what I was "bad" at. The argument went on to say that this was perhaps one reason why females were more likely to choose to go into liberal arts than STEM fields.
EDIT: As Vladimir Nesov points out below, don't just study what I recommend; this is a list of things that you might look up outside of classes to help stay motivated while doing a degree in mathematics, more than a list of things you should study to learn mathematics outside of school.
Also refined point (7).
I am a mathematics graduate student; I currently focus on number theory and arithmetic geometry. So here are a few areas I'd recommend, coming from different goal structures I have:
1) if you are interested in learning things that are really cool and beautiful I would recommend elementary number theory, for example from Hatcher's visual approach. This doesn't require much heavy grounding and is absolutely awesome and has neat pictures. If you want to continue on this path, p-adic numbers are the place to go.
2) if you are interested in studying advanced mathematics, I'd recommend studying representation theory and category theory; these seem to have lots of applications in almost every area of mathematics, including algebra, number theory, mechanics, geometry, and topology. Maybe less in analysis or logic, although more logicians' perspectives on category theory seems valuable to me. Also complex analysis.
3) If you are interested in abstract concepts that feel like they have universal applicability (I don't know how much the metaphors I draw from these actually help me but I draw them almost constantly): linear algebra, group theory, and basic real analysis. Symmetry and distance are every day concepts; seeing them mathematically derived was very cool to me.
4) if you want to make lots of money, I think calculus and dynamical systems lead most directly into financial modeling; I'm not really sure.
5) if you want to do interesting research with real life applications, you might be better sticking to statistics and probability theory; although dynamical systems have their applications in game theory the impression that I get is that the difficulty mostly comes from differential equations and computation complexity, not from mathematical insights.
6) You probably shouldn't study algebraic geometry. I do a little, but it is filled with technical definitions and complex terminology and it has a reputation for taking people a very long time to be able to understand at all. If you want an intellectual challenge for yourself maybe it's appropriate, but if you want to learn a field quickly and use the insights it is probably more trouble than it is worth, at least until some amazing new text book comes out on it which I doubt may ever happen. It is too late for me, save yourself!
7) if you are in school at a university, I'd suggest looking up math professors on ratemyprofessors, the ratings aren't perfect but it does look like they correlate well with my experiences with my professors. Requiring slightly more effort but giving much better information would be asking other math majors or TAs about different professors' teaching styles. And then, just take courses from good professors. This is probably worthwhile in any subject; better professors are going to mean more than good classes. Taking a class with a good professor means you will probably enjoy the class, taking it with a bad professor means you probably won't. I don't think this is the context of your question but it is probably relevant to others asking similar questions.
I don't believe specialization on the level you imply is sustainable. You'd get lost even on upper undergraduate level if all topics outside most of your recommended sets are completely left out. The mathematical maturity that allows you to imagine limiting your study to just a few topics came from having studied the others.
I probably didn't make this clear, but I do agree with you. If you want to study mathematics you need to study lots of areas which is why they have general requirements like that for getting degrees in mathematics.
But if you are an undergraduate looking for a particular subject to get you in the "mathematics groove" I think (1) is a good recommendation for independent study alongside classes, (3) is really a way to lay a foundation for mathematical maturity; certainly a class on proofs and logic would be necessary before almost any of my recommendations.
Again: I didn't mean my list to be a list of "only study these things" so much as "if you want more math, these might be good places to look."
I'd suggest looking up math professors on , the ratings aren't perfect but it does look like they correlate well with my experiences with my professors.
Did you mean to link to Rate My Professor?
On a lark I looked at my favorite professors (that is, the ones I felt taught me the most) and all of them have ratings below 3.0.
I think RMP is more for undergrads who want to coast through their degree with the least amount of effort. (This makes a certain amount of sense.)
My favorite professors are all rated 4.5 or higher, and my least favorites are rated 3.0 or lower; ones I'd rate in between are commonly rated higher than my favorites, but it contains substantially greater than 0 information relative to my school. Apparently YMMV!
I would guess talking to TAs or other same-major students is going to get most LWers a more useful perspective than RMP, but website reviews are still better than nothing.
website reviews are still better than nothing.
Only detailed reviews. If you don't know what the criteria are, they can easily be worse - counting as negative a factor you would consider positive, for example.
I would suggest that if you know nothing about the rating system, it is still likely to positively correlate because of universal factors like speaking clearly. In the case of RMP, I'd suggest that you'd expect an even better correlation because easiness and attractiveness are asked about separately. It's still possible for this not to work out because of what you suggest, but it seems less likely to me on average.
I think RMP is more for undergrads who want to coast through their degree with the least amount of effort.
RMP is also very useful if you want to sign-up for classes taught by attractive professors. I like to have something nice to look at while not doing course-work, lol!
I agree with you, I was just trying to help magfrump.
the utility of different mathematical concentrations
The answer is highly dependent on one's goals, interests, and personality. Hence there is unlikely to be any particular document which successfully explains the answer in a way that applies to everyone.
What would be better would be for people interested in learning mathematics to post comments explaining the nature of their own particular interest, and to receive individually-tailored replies. If general patterns emerge from those replies, these general patterns can then be extracted and abstracted afterward, preferably with appropriate cautions and disclaimers.
Alternatively or in addition, those who already use some field of mathematics in their daily work might post examples of what they need to know. For example, in physics research I use (off the top of my head) a bit of calculus, considerable statistics, occasional trig, and large amounts of plain symbol-manipulating algebra. The most relevant math skill, perhaps, is that difficult-to-teach intuition, the fingertip-feel of how to manipulate an equation so you get it in a form you can use.
Agreed. Advice is trivial to give, especially in the abstract, and therefore likely to have a low signal-to-noise ratio. Extrapolating patterns after dealing with specific instances will probably increase the SNR substantially.
I am interested in Computer Science, compiler optimization, and machine learning but I know relatively little about these subjects.
I am currently reading Concrete Mathematics.
Don't forget to learn some functional analysis, especially Hilbert spaces and how they're used in, for example, quantum theory and stochastic processes.
Here's one such resource: The simple math of everything
What resources exist detailing which mathematics to learn in what order?
One could acquire a university syllabus for a degree in mathematics and use textbooks and or MIT Opencoursware to learn with.
What resources exist that explain the utility of different mathematical concentrations for the purpose of directing studies?
I am having trouble understanding this question. By the words "mathematical concentrations", could one substitute the term "mathematical subfields"? If so, then I would say the best resource could be more experienced people with similar instrumental goals to one's self.
I am having trouble understanding this question. By the words "mathematical concentrations", could one substitute the term "mathematical subfields"
Yes, thank you, I performed the substitution in the original post.
If so, then I would say the best resource could be more experienced people with similar instrumental goals to one's self.
Which is precisely why I am asking LessWrong.
Here's one such resource: The simple math of everything
EDIT: As Vladimir Nesov points out below, don't just study what I recommend; this is a list of things that you might look up outside of classes to help stay motivated while doing a degree in mathematics, more than a list of things you should study to learn mathematics outside of school.
Also refined point (7).
I am a mathematics graduate student; I currently focus on number theory and arithmetic geometry. So here are a few areas I'd recommend, coming from different goal structures I have:
1) if you are interested in learning things that are really cool and beautiful I would recommend elementary number theory, for example from Hatcher's visual approach. This doesn't require much heavy grounding and is absolutely awesome and has neat pictures. If you want to continue on this path, p-adic numbers are the place to go.
2) if you are interested in studying advanced mathematics, I'd recommend studying representation theory and category theory; these seem to have lots of applications in almost every area of mathematics, including algebra, number theory, mechanics, geometry, and topology. Maybe less in analysis or logic, although more logicians' perspectives on category theory seems valuable to me. Also complex analysis.
3) If you are interested in abstract concepts that feel like they have universal applicability (I don't know how much the metaphors I draw from these actually help me but I draw them almost constantly): linear algebra, group theory, and basic real analysis. Symmetry and distance are every day concepts; seeing them mathematically derived was very cool to me.
4) if you want to make lots of money, I think calculus and dynamical systems lead most directly into financial modeling; I'm not really sure.
5) if you want to do interesting research with real life applications, you might be better sticking to statistics and probability theory; although dynamical systems have their applications in game theory the impression that I get is that the difficulty mostly comes from differential equations and computation complexity, not from mathematical insights.
6) You probably shouldn't study algebraic geometry. I do a little, but it is filled with technical definitions and complex terminology and it has a reputation for taking people a very long time to be able to understand at all. If you want an intellectual challenge for yourself maybe it's appropriate, but if you want to learn a field quickly and use the insights it is probably more trouble than it is worth, at least until some amazing new text book comes out on it which I doubt may ever happen. It is too late for me, save yourself!
7) if you are in school at a university, I'd suggest looking up math professors on ratemyprofessors, the ratings aren't perfect but it does look like they correlate well with my experiences with my professors. Requiring slightly more effort but giving much better information would be asking other math majors or TAs about different professors' teaching styles. And then, just take courses from good professors. This is probably worthwhile in any subject; better professors are going to mean more than good classes. Taking a class with a good professor means you will probably enjoy the class, taking it with a bad professor means you probably won't. I don't think this is the context of your question but it is probably relevant to others asking similar questions.
I don't believe specialization on the level you imply is sustainable. You'd get lost even on upper undergraduate level if all topics outside most of your recommended sets are completely left out. The mathematical maturity that allows you to imagine limiting your study to just a few topics came from having studied the others.